By Mark Ryan

If you want to work with multiple-plane proofs, you first have to know how to determine a plane. Determining a plane is the fancy, mathematical way of saying “showing you where a plane is.”

There are four ways to determine a plane:

  • Three non-collinear points determine a plane. This statement means that if you have three points not on one line, then only one specific plane can go through those points. The plane is determined by the three points because the points show you exactly where the plane is.

To see how this works, hold your thumb, forefinger, and middle finger so that your three fingertips make a triangle. Then take something flat like a hardcover book and place it so that it touches your three fingertips. There’s only one way you can tilt the book so that it touches all three fingers. Your three non-collinear fingertips determine the plane of the book.

  • A line and a point not on the line determine a plane. Hold a pencil in your left hand so that it’s pointing away from you, and hold your right forefinger (pointing upward) off to the side of the pencil. There’s only one place something flat can be placed so that it lies along the pencil and touches your fingertip.
  • Two intersecting lines determine a plane. If you hold two pencils so that they cross each other, there’s only one place a flat plane can be placed so that it rests on both pencils.
  • Two parallel lines determine a plane. Hold two pencils so that they’re parallel. There’s only one position in which a plane can rest on both pencils.